CHAPTER 1 Introduction
MULTIPLE CHOICE
Sections 1.1-1.7
1. Under descriptive statistics, we study:
a. the description of decision making tricks
b. the methods for organizing, displaying, and describing data
c. how to describe the probability distribution
d. samples to assist in decision making
2 Under inferential statistics, we study:
a. the methods to make decisions about population based on sample results
b. how to make decisions about mean, median, or mode
c. how a sample is taken from a population
d. tables composed of summary measures
3 In statistics, a population consists of:
a. all people living in a country
b. all people living in the area under study
c. all subjects or objects whose characteristics are being studied
d. a selection of a limited number of elements
4 In statistics, we define a sample as:
a. people living in one city only
b. the target population
c. all items under investigation
d. a portion of the population
5. In statistics, conducting a survey means:
a. collecting information from elements
b. making mathematical calculations
c. drawing graphs and pictures
d. none of these
6. In statistics, conducting a census means:
a. making decisions based on sample results
b. checking if a variable is qualitative or quantitative
c. collecting information from all members of the population
d. collecting a sample with replacement
7. In statistics, a representative sample is the sample that:
Introductory Statistics, Sixth Edition
a. contains the characteristics of the population as closely as possible
b. represents the results of a sample exactly
c. contains all people living in an area
d. contains elements collected with replacement
8. A random sample is a sample drawn in such a way that:
a. each member of the population has .10 chance for being included in the sample
b. all elements of a population are included
c. some members of the population have no chance of being included in the sample
d. each member of the population has some chance for being included in the sample
9 A simple random sample is a sample drawn in such a way that:
a. each member of the population has some chance for being included in the sample
b. every tenth element of an arranged population is included
c. each member of the population has equal chance for being included in the sample
d. each member of the population has .10 chance for being included in the sample
10 A data set is a:
a. set of decisions made about the population
b. set of graphs and pictures
c. collection of observations on a variable
d. score collected from an element of the population
11 A variable is a:
a. result that varies from sample to sample
b. characteristic under study that assumes different values for different elements
c. sample drawn from a population in such a way that all its elements are different
d. characteristic under study that assumes the different values for identical elements
12. An observation is the:
a. graph observed for a data set
b. value of a variable for a single element
c. table prepared for a data set
d. sample observed from the population
13. A quantitative variable is the one that can:
a. assume numerical values
b. be graphed
c. be used to prepare tables
d. have no intermediate values
14. A discrete variable is a variable that can assume:
a. categorical values only
b. countable values only
c. uncountable values
d. non-numerical values
15 A continuous variable is a variable that can assume:
a. categorical values only
b. countable values only
c. uncountable values
d. non-numerical values
Chapter 1 Introduction
16. A qualitative variable is the one that:
a. can assume numerical values
b. cannot be graphed
c. can assume uncountable values
d. cannot be measured numerically
17. Time-series data are collected:
a. on the same element for the same variable for different periods of time
b. on a variable that involves time, e.g., minutes, hours, weeks, months, etc.
c. for a qualitative variable
d. on different elements for the same period of time
18 Cross-section data are collected:
a. on the same element for the same variable for different periods of time
b. on different elements for the same period of time
c. for a qualitative variable
d. on different elements for the same variable for different periods of time
Section 1.8
Use the following information for questions 19 – 22:
The telephone bills for the past month for four families are $45, $69, $43, and $76.
19 The value of x is:
a. 243
b. 223
c. 233
d. 278
20 The value of x2 is:
a. 50,625
b. 12,768
c. 14,411
d. 13,561
21 The value of (x)2 is:
a. 54,289
b. 49,729
c. 25,930
d. 32,768
22. The value of (x – 5) is:
a. 203
b. 225
c. 221
d. 213
Use the following information for questions 23 – 26:
The test scores of five students are 87, 66, 98, 74, and 90.
23. The value of x is:
a. 415
b. 389
c. 400
d. 417
24 The value of x2 is:
a. 166,464
b. 35,105
c. 34,230
d. 49,829
25 The value of (x)2 is:
a. 168,100
b. 33,894
c. 187,683
d. 172,225
26 The value of (x – 10) is:
a. 365
b. 395
c. 360
d. 429
Use the following information for questions 27 – 33:
Consider the following five pairs of m and f values:
27 The value of m is:
a. 68
b. 41
c. 37
d. 40
28 The value of mf is:
a. 163
b. 157
c. 153
d. 212
29 The value of m2 is:
a. 289
b. 336
c. 671
d. 359
Chapter 1 Introduction
Introductory Statistics, Sixth Edition
30. The value of f2 is:
a. 73
b. 65
c. 75
d. 90
31. The value of m2f is:
a. 751
b. 153
c. 1,546
d. 820
32. The value of mf2 is:
a. 751
b. 153
c. 1,546
d. 820
33 The value of (m – 3)2f is:
a. 646
b. 1,290
c. 694
d. 1,095
Use the following information for questions 34 – 40:
Consider the following six pairs of x and y values:
34 The value of y is:
a. 102
b. 133
c. 106
d. 110
35 The value of xy is:
a. 1,046
b. 1,643
c. 1,655
d. 1,813
36 The value of xy2 is:
a. 12,675
b. 37,817
c. 39,365
d. 27,548
37 The value of x2y is:
a. 23,478
b. 14,233
c. 27,367
d. 29,431
38. The value of x2 is:
a. 1,143
b. 891
c. 1,303
d. 2,190
39. The value of y2 is:
a. 1,143
b. 891
c. 1,303
d. 2,190
40. The value of (x – 2)2y is:
a. 19,786
b. 10,370
c. 23,235
d. 47,925
Chapter Review Questions
41. Given the following data:
Which one of the following statements is true?
a. x = 20
b. y = 2
c. xy = 41
d. y2 = 45
42. Which one of the following is an example of qualitative data?
a. total quarterly revenue
b. average graduation rate for athletes
c. number of visitors to theme parks last year
d. For major colleges and universities, we record whether or not their enrollment increased from last year to this year.
43. Which one of the following is an example of cross-section data?
a. daily checking account balance last year
b. total insect population among 12 U.S. national parks in 2003
c. consumption (in pounds) of pork products in the U.S. from 1983 to 2003
d. total value of U.S. Savings Bonds each year from 1994 to 2004
44. Which one of the following is a continuous variable?
a. types of boats owned by the U.S. Coast Guard
b. lengths of top-ten hit songs
c. number of candles lit during the Christmas season
d. the outcome of a single coin flip
45. A statistician wants to determine the average annual Gross National Product for countries in Africa. He samples the 20 largest African countries over 10 years, and gets their quarterly GNP results for each quarter of each year. Which one of the following is a true statement concerning the statistician’s study?
a. Algeria, Libya, and Egypt are possible elements of the study.
b. The data he collects is cross-section data, and there is no time-series data collected.
c. The population is all people in Africa.
d. The sample is a representative sample.
46 A statistician wants to determine the total annual medical costs incurred by all U.S. states from 1981 to 2001 as a result of health problems related to smoking. She polls each of the fifty states annually to obtain health care expenditures, in dollars, on smoking-related illnesses. Which one of the following is not a true statement?
a. The population is all U.S. states.
b. The sample is the same as the population.
c. The data collected are qualitative data.
d. You can consider each state as an element of the study.
47 A scientist is conducting an experiment to determine the relationship between consumption of a certain type of food and high-blood pressure. He conducts a random sample on 2,000 people and first asks them a “yes” or “no” question: Do you eat this type of food more than once a week? He also takes the blood pressure of each person and records it (example: 120/80). Which one of the following statements is true?
a. The blood-pressure data is quantitative data.
b. The data regarding whether each person eats the food regularly is quantitative data.
c. The population is all U.S. males.
d. The data collected is time-series data.
48. As part of a study on worker productivity, you ask three employees (John, Mary, and Chris) at the end of a particular work week to estimate how long, in minutes, their morning coffee breaks were on each day of that week. Which one of the following lists describes the elements, or members, of the sample?
a. Monday, Tuesday, Wednesday, Thursday, Friday
b. 14.5 minutes
c. 15 minutes, 12 minutes, 11 minutes, 14 minutes, 16 minutes
d. John, Mary, Chris
49. Given the following data:
Which one of the following statements is not true?
a. y2 = 8
b. yz = –1
c. xy2 = 34
d. xyz = –7
50. A research team conducts an experiment on an automotive plant to determine the effect of production demands on stress levels. Which one of the following describes the drawing of a random sample of all paid employees of the plant?
a. The team selects names at random from the payroll for the sample.
b. The team selects for the sample the first 20 employees who enter the building after 8:00 a. m..
c. The team selects for the sample all machine operators.
d. The team puts the names of all employees with more than 6 months’ experience at their present job into a hat and draws 20 names at random from the hat for the sample.
Multiple Choice Summary Table
PROBLEM AND ESSAY
Sections 1.1-1.7
1 Explain the two meanings of the word “statistics.”
2 Briefly describe the areas of statistics.
3. Explain the difference between a population and a sample.
4. Explain which of the following constitute a population and which constitute a sample.
a. Incomes of 500 families selected from New York
b. Salaries of all employees of a company
c. Number of absences during the semester for each of the students in a class
d. Number of cars owned by each of the 100 families selected from a city
e. Color of hair of 25 girls
5 Describe the difference between a sample survey and a census.
6 What is a representative sample?
7 Explain the difference between the sampling with replacement and the sampling without replacement. Give one example of each.
8 List several reasons why a sample survey is preferable to conducting a census.
9 Define the following terms and give an example of each: a variable, an observation, and a data set.
10 The following table gives the names and ages of five persons.
Name Age Bill 39 Dawn 27 Sharmon 23 Joe 31 Connie 21
Explain the meaning of member, variable, measurement, and data set with reference to this table.
11. The following table gives the number of employees working for each of the four companies.
Company Employees
Random Corporation 385 Silver Manufacturers 291 Bootstrap Company 725 White Paper Inc. 569
Explain the meaning of member, variable, measurement, and data set with reference to this table.
12 Briefly describe the difference between a quantitative and a qualitative variable. Explain what is meant by discrete and continuous variables. Give one example of each of such variables.
13 Which of the following variables are quantitative and which are qualitative? Classify the quantitative variables as discrete or continuous.
a. Weight of a package
b. Color of eyes of people
c. Price of a concert ticket
d. Number of televisions owned by families
e. Number of births on a day in a hospital
14. Which of the following variables are quantitative and which are qualitative? Classify the quantitative variables as discrete or continuous.
a. Model of a car
b. Food expenditure per month for families
c. Number of children per family
d. Commuting time from home to work for a person
e. Number of houses on a block
15 Explain the difference between cross-section and time-series data. Give one example of each of these two types of data.
16 Classify the following as cross-section or time-series data.
a. Heights of 30 basketball players
b. Total student enrollment at a college during 1993-2003
c. Number of employees working for a company at the end of each of the past 10 months
d. Mortgage paid by 25 homeowners
17 Classify the following as cross-section or time-series data.
a. Entertainment expenditure incurred by a family during each of the past 12 months
b. Time taken by 20 students to complete a test
c. Prices of 50 houses
d. Prime interest rate during each of the past 20 months
Section 1.8
18 The ages of five employees of a company are 47, 28, 55, 41, and 52 years. Find:
a. x
b. (x – 6)
c. (x)2
d. x2
19 The heights of six persons are 65, 69, 72, 62, 74, and 67 inches. Find:
a. x
b. (x – 10)
c. (x)2
d. x2
20 Consider the following six pairs of m and f values.
Compute the following:
a. m e. f2
b. f f. m2f
c. mf g. mf2
d. m2 h. (m – 5)2f
21. Consider the following seven pairs of x and y values.
Compute the following:
a. x e. y2
b. y f. x2y
c. xy g. xy2 d. x2 h. (x – 3)2y
Chapter Review Questions
22 A brokerage firm conducts a survey to try to predict the future stock performance of a particular company. The following data are collected:
Calculate y and y2. Is this cross-section or time-series data? Briefly explain.
23. You select 45 students from a class of 100 at random and ask them if they enjoy their math classes. You next enter the name and response of each of the students into a database for future reference. Identify the population, the sample, the set of elements, and the variables. Will the data collected be quantitative or qualitative? Briefly explain.
24. An engineering student is concerned about how much time she should spend studying for her upcoming engineering certification exam. She contacts 50 recent graduates of her school’s engineering program and asks each of them two questions:
1) Did you pass your certification exam?
2) How much time did you spend preparing for the exam?
She uses this information to schedule her study time in the two months before her exam. Would you consider her method to be descriptive or inferential? Briefly explain.
25 Twenty meteorologists participate in a study. Each person predicts the daily high temperature for the next seven days. An accurate prediction is one that falls within 2 F of the actual temperature. The researcher records values of “accurate” or “inaccurate” at the end of each day for each person. At the end of the seven days, the researcher calculates each person’s total accuracy by averaging his/her predictions over the seven days and compares this number with
the actual average temperature. Identify the quantitative and qualitative variables, and whether the quantitative data is discrete or continuous.
26. You record daily values of a variable x over sixteen days. What is the summation notation for the average of these values?
27. You want to find out what percentage of citizens over 65 in your town still work full-time. Starting from the front of the telephone book, you call each number, ask whether the person is over 65 and, if so, whether he or she works full-time. You call the numbers in order as they appear in the phone book until you have obtained data on fifty citizens over 65, or until you have reached the end of the phone book, whichever comes first. Identify the population. Have you obtained a random sample of the population? A simple random sample? Briefly explain.
28 Your insurance company has collected data to determine the performance of its life insurance products over the past calendar year. Unfortunately, the data for November was misplaced. The data for the other months appear in the following table:
Month Policies Sold Month Policies Sold
Jan 12 Jul 10
Feb 15 Aug 10
Mar 20 Sep 20
Apr 15 Oct 14
May 6 Nov ?
Jun 18 Dec 9
Suppose the company sold 160 policies in the past year. How many policies did it sell in November?
29. We generally consider scores on a test to be quantitative data because they usually represent numerical data. In what way might we consider test scores qualitative data?
30. An agency collects data on murder rates from six U.S. cities in 2003, 2004, and 2005. In what way is this cross-section data? In what way is it time-series data?
31. Fund managers at a particular investment firm are considered successful if the funds that they manage increase faster than the Standard and Rich’s 400 stock index. Joe Fundmanager’s boss is evaluating him based on the performance of his fund over the last six months as compared to the S&R 400. Six months ago his fund was worth $2,000,000. The following table shows the monthly increase in value of Joe’s fund. The table also shows how that the net worth would have increased in value had Joe invested equally among the 400 companies in the S&R:
Develop and use a statistic to evaluate the performance of Joe’s fund. Can you conclude that Joe is a successful fund manager?
Problem and Essay Solutions
1. The word “statistics” could be referring to numbers or numerical facts, but it could also be referring to the field or discipline of study that surrounds the methods used to collect, analyze, present, and interpret data for decision making.
2. The two major areas of statistics are descriptive and inferential. We use descriptive statistics to organize, display, and describe data by using table, graphs, and summary measures. Inferential statistics use sample results to assist in making predictions or drawing conclusions about a population.
3. A population represents the entire collection of units about which information is sought, whereas a sample is a smaller portion of the population from which measurements are taken in the course of an investigation.
4 a. Sample
b. Population
c. Population
d. Sample
e. Sample
5 A sample survey seeks to obtain information from a smaller part (sample) of the population, whereas we conduct a census to obtain information from the entire population.
6 A representative sample is a mini version of the population. It contains all the key features found in the population and is a fair reflection of the reality that we would see in the population.
7 When you build a sample without replacement, you extract elements of the population and do not return them to the population as you add additional elements to the sample. The size of the population shrinks as the sample enlarges, and it is impossible for the sample to include two identical elements. On the other hand, sampling with replacement requires that the size of the population remain constant as the sampling proceeds. In this case you extract a sample element, measure and record its properties, and then return the sample element to the population before extracting another element. Thus the size of the population remains constant as the sample enlarges, and it is possible for the sample to include measurements stemming from two identical elements.
8. a. Too much time and/or money might be required to study the entire population. b. It may be impossible to identify and secure all the members of the population. c. The testing or measuring process may be destructive, and testing the entire population may damage all the members and leave nothing behind that you can salvage and use.
9. A variable is a characteristic that can take on different values for different elements of the population. The value of a variable for one element of the population is an observation. The data set represents the collection of all the observations made on one or more variables.
10. Member: a person listed in the table
Variable: age
Measurement: the age of a particular person
Data set: the ages of all five persons
11. Member: a company listed in the table
Variable: the number of employees
Measurement: the number of employees for a particular company
Data set: the number of employees for all five companies
12 Qualitative data describe a particular characteristic of an item and are usually categorical in nature, whereas quantitative data are numerical and usually involve counting or enumerating. Discrete data involve counting or enumerating, and the only possible values are positive integers, whereas data that are continuous can take on any one of an infinite number of possible values over an interval on the number line. The number of parking tickets that people receive each year is discrete. The heights and weights of people is continuous.
13 a. Quantitative, continuous d. Quantitative, discrete b. Qualitative, continuous e. Quantitative, discrete c. Quantitative, continuous
14 a. Qualitative, discrete d. Quantitative, continuous b. Quantitative, continuous e. Quantitative, discrete c. Quantitative, discrete
15 Time series data are observations collected at regular time intervals, such as yearly, monthly, weekly, daily, etc. We collect cross-sectional data within a given period of time that may even be instantaneous and not in a predefined interval or sequence.
16 a. Cross-section data c. Time-series data b. Time-series data d. Cross-section data
17. a. Time-series data c. Cross-section data b. Cross-section data d. Time-series data
18. a. 47 + 28 + 55 + 41 + 52 = 223
b. 41 + 22 + 49 + 35 + 46 = 193
c. (223)2 = 49,729
d. 2,209 + 784 + 3,025 + 1,681 = 10,403
19. a. 65 + 69 + 72 + 62 + 74 + 67 = 409
b. 55 + 59 + 62 + 52 + 64 + 57 = 349
c. (409)2 = 167,281
d. 4,225 + 4,761 + 5,184 + 3,844 + 5,476 + 4,489 = 27,979
20. a. 15 + 25 + 12 + 16 + 11 + 30 = 109
b. 10 + 13 + 7 + 3 + 5 + 16 = 54
c. 150 + 325 + 84 + 48 + 55 + 480 = 1,142
d. 225 + 625 + 144 + 256 + 121 + 900 = 2,271
e. 100 + 169 + 49 + 9 + 25 + 256 = 608
f. 2,250 + 8,125 + 1,008 + 768 + 605 + 14,400 = 27,156
g. 1,500 + 4,225 + 588 + 144 + 275 + 7,680 = 14,412
h. 1,000 + 5,200 + 343 + 363 + 180 + 10,000 = 17,086
Introductory Statistics, Sixth Edition
21. a. 8 + 11 + 16 + 7 + 9 + 13 + 12 = 76
b. 4 + 8 + 10 + 5 + 6 + 3 + 2 = 38
c. 32 + 88 + 160 + 35 + 54 + 39 + 24 = 432
d. 64 + 121 + 256 + 49 + 81 + 169 + 144 = 884
e. 16 + 64 + 100 + 25 + 36 + 9 + 4 = 254
f. 256 + 968 + 2,560 + 245 + 486 + 507 + 288 = 5,310
g. 128 + 704 + 1,600 + 175 + 324 + 117 + 48 = 3,096
h. 100 + 512 + 1,690 + 80 + 216 + 300 + 162 = 3,060
22 y = 115.5, y2 =. 1907.125 This is time-series data because values exist for the same variable and for the same element at different points in time.
23 Population: 100 students in the class
Sample: 45 randomly selected students from the class
Set of elements: the names of the 45 students selected Variables: response, which could be “yes” or “no”, and possibly “maybe”
The data will be qualitative because we can group the responses into two (or possibly three) categories.
24 Inferential, since she is using the data collected to make a decision.
25 Continuous and quantitative: average predicted temperature for each person Qualitative: daily values of “accurate” or “inaccurate”
26 x/16
27 Population: citizens over 65 in your town
This is a random sample because everyone has some chance of being called. However, it is not a simple random sample since those with last names starting with “A” are much more likely to be called than those with last names ending in “Z”.
28. 160 – 12 – 15 – 20 – 15 – 6 – 18 – 10 – 10 – 20 – 14 – 9 = 11 policies
29. If the data is collected as “pass” or “fail” without specific regard for the actual score, it is qualitative. More generally, one might consider whether the test score was above or below a specified level.
30. The data is cross-section data in that it is collected from six different cities at the same time every year. It is time-series data in that it is collected from the same six cities over a number of years.
31. y would be the statistic used for Joe’s fund, and would be evaluated against z. In this case, y = $110,000 and z = $113,000. Therefore, Joe is not a successful manager.